Problem 4. Let X = (X₁, X2,..., Xn) be i.i.d. where each X, has proba-bility mass functionP(Xi = x₁) =0(1-0)¹-il, a; € (-1,0,1), 0≤0≤1.a. Derive the MLE Ô for 0.b. Assuming n is large, find the approximate distribution of .c. Find an approximate 95% confidence interval for 0.d. Derive the likelihood ratio test for testing 0 = 1/2 vs. 0 > 1/2.

a. To find the maximum likelihood estimator (MLE) of theta, we need to maximize the likelihood…

Answered: Problem 4. Let X = (X₁, X2,..., Xn) be…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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1. (a) Suppose that x1 = -5, x2 Calculate the following quantities: 36, х3 — 4, and yi %3D —7, У2 — 0.5, уз — 1. 2 3 3 ii. iii. lyı|+ i=2 i=1 i=2 (b) Classify each (continuous) or categorical. If a variable is categorical, further classify it as either nominal or ordinal. Justify your answer. of the following variables either measurable one as i. Exchange rate between two currencies. ii. Candidate number in an examination. iii. University degree type (in terms of Bachelors, Masters, Ph.D.). (c) State whether the following statements are true or false, and provide a brief explanation. i. For a set of observations x1, x2, . .. , xn, with mean ĩ, then: (x; – ) P(A) + P(B). iii. For a random variable X, (E(X))² can be greater than E(X?). iv. Failure to reject a false null hypothesis is known as the power of a test. v. A 5-by-3 contingency table which results in a x? test statistic value of 15.312 is statistically significant at the 1% significance level.
Qn3. Suppose that for a certain life the force of mortality is defined as H (x) -,0
A fish population in a lake satisfies dB/dt=cB(K-B) Fish are then added at A fish per hour. Find the DE for B with respect to t. Is there a new limiting population? If so, what is the solution?
6. (a) Consider a fish population, assume that the total number of fish is constant. Some of fish live in a river, some live in a lake that the river flows into. Let R(t) and L(t) denote the fish populations in the river and in the lake respectively, after t months. Suppose each month, 10% of the fish in the river move into the lake and 5% of the fish in the lake move to the river. Set up a system of differential equations for the rates of change of R(t) and L(t). Do not solve.
Suppose f, f', g, and g' have the following values: 1 2 3 4 f(x) 8 2 1 -8 f'(x) g(x) 10 -7 - 2 6 -6 – 253 - 252 - 246 - 248 - 262 - 64 56 14 - 15 23 (2),6 At the half-time show for a Milwaukee College football game, a balloon will travel along a path with equation z + y = f(x)g(x) and will be released at the point where x = 2. To ensure the safety of the crowd, Sophia needs the equation of the line tangent to the graph of 2° + y° = f(x)g(x) at x = 2 For this question, if you choose to use decimals, be sure that you round your numbers to at least 4 decimal places. The equation of the tangent line is:
Ex. 2: Consider the process X (t) = 10 sin (200t + 0) where o is uniformly distributed in the interval (- n , n). Check whether the process is stationary or not. %3|
Problem 46. Let I = [a,b] be a closed interval and let p e I. Is p always an accumulation point of I? Prove your answer is correct.
1. If y = f(x) is a continuous function for all 0 sx< 6, and f(0) = 1 and (6) = 7, then which of the following could be false: 0. 6. (a) f(x) has no vertical asymptotes on 0
8-9. If f(z, y) 3 теу + х?у3 find fa = fy = fyy %| fay ||

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Problem 4. Let X = (X₁, X2,..., Xn) be i.i.d. where each X, has proba- bility mass function P(X = ) - (9) (1-0)¹-1, ₁ € (-1,0,1}, 0≤ 0≤1.